Definition of the Stochastic Integral

Abstract

In this chapter, we shall define stochastic integrals of the form $int_{[0,t]}$ X dM where M is a right continuous local L ²-martingale and X is a process satisfying certain measurability and integrability assumptions, such that the family of stochastic integrals $\{ \int_{{[0,t]}} {X\,dM,t \in {{\mathbb{R}}_{ + }}} \}$ is a right continuous local L ²-martingale. For certain M and X, the integral can be defined path-by-path. For instance, if M is a right continuous local L ²-martingale whose paths are locally of bounded variation, and X is a continuous adapted process, then $int_{[0,t]}^{}{{X_s}d{M_s}}(\omega )$ is well-defined as a Riemann-Stieltjes integral for each t and ω, namely by the limit as n → ∞ of

$$ \sum\limits_{{k = 0}}^{{[{{2}^{n}}t]}} {{{X}_{{k{{2}^{{ - n}}}}}}} (\omega )({{M}_{{(k + 1){{2}^{{ - n}}}}}}(\omega ) - {{M}_{{k{{2}^{{ - n}}}}}}(\omega )). $$